You Do Need power plug adapter while traveling from North Cyprus to Cayman Islands.

Here is Why You Need An Adapter

The outlet and plug type used in Cayman Islands are A and B and in North Cyprus is G. This is how they look like.

North Cyprus

In North Cyprus, outlets of type G are in use.

Type G

Cayman Islands

In Cayman Islands, outlets of type A and B are in use.

Type A

Type B